Scaling properties of a structure intermediate between quasiperiodic and random |
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Authors: | S. Aubry C. Godrèche J. M. Luck |
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Affiliation: | (1) Institute for Theoretical Physics, University of California, 93106 Santa Barbara, California;(2) Laboratoire Léon Brillouin, Laboratoire Associé CEA-CNRS, France;(3) SPSRM: Service de Physique du Solide et de Résonance Magnétique, CEN Saclay, 91191 Gif-sur-Yvette, France;(4) SPhT: Service de Physique Théorique, CEN Saclay, 91191 Gif-sur-Yvette, France |
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Abstract: | We consider a one-dimensional structure obtained by stringing two types of beads (short and long bonds) on a line according to a quasiperiodic rule. This model exhibits a new kind of order, intermediate between quasiperiodic and random, with a singular continuous Fourier transform (i.e., neither Dirac peaks nor a smooth structure factor). By means of an exact renormalization transformation acting on the two-parameter family of circle maps that defines the model, we study in a quantitative way the local scaling properties of its Fourier spectrum. We show that it exhibits power-law singularities around a dense set of wavevectorsq, with a local exponent(q) varying continuously with the ratio of both bond lengths. Our construction also sheds some new light on the interplay between three characteristic properties of deterministic structures, namely: (1) a bounded fluctuation of the atomic positions with respect to their average lattice; (2) a quasiperiodic Fourier transform, i.e., made of Dirac peaks; and (3) for sequences generated by a substitution, the number-theoretic properties of the eigenvalue spectrum of the substitution. |
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Keywords: | Quasiperiodicity inflation rules circle maps self-similarity |
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